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(levels of the independent variable) in the design. In our SAT study time
example with five groups, we can make no more than four orthogonal
comparisons.
Comparisons that are not orthogonal to each other are not indepen-
dent of each other. These nonorthogonal comparisons subsume the full
range of the remaining pairwise and nonpairwise comparisons. Because
we can infinitely adjust the weights in the weighted linear combinations
of composite comparisons, it is possible to conceive of an infinite number
of nonorthogonal comparisons (Kirk, 1995).
When you make comparisons that are not orthogonal to each other,
you are doing redundant or overlapping work. In fact, all of these non-
orthogonal comparisons “ . . . can be expressed as a linear combination
of [the orthogonal] contrasts” (Kirk, 1995, p. 115). However, in the end,
research interest and not orthogonality is the more important consid-
eration in determining which comparisons to evaluate (Myers & Well,
1991). In this spirit, Toothaker (1993) has argued that the issue of orthog-
onality may be one of the less important ones in the context of multiple
comparison procedures. We therefore bring the issue of orthogonality of
comparisons to your attention so that you understand it, but we recom-
mend following the lead of Myers and Well (1991) in performing those
comparisons that are most relevant to your research whether or not they
are orthogonal.
7.5 STATISTICAL POWER
We discussed the idea of statistical power in Chapter 4. Power reflects the
ability to find an effect when one exists. Researchers gain more power
when their sample size is larger, when the population effect size is larger,
and when their alpha level is less stringent. Power comes into play in the
context of multiple comparison procedures. It is very common in the
research literature and in statistics textbooks for the authors to speak of
andtoselectmultiplecomparisonproceduresonthebasisofpower.
7.5.1 THE FOCUS OF THE POWER
When we speak of power, the focus is on the mean differences we observe.
In our SAT preparation example, we have five means and have obtained a
significant F ratio in our omnibus analysis. When we examine our mean
differences, they will all be of somewhat different magnitudes. It is these
differences that are the focus when we speak of the power of a particular
test.
7.5.2 APPLYING THE CONCEPT OF POWER
In an absolute sense, we do not know - nor can we ever know - which
means are truly different from which others. Any multiple comparison
procedure we use to deal with the question of group mean differences
gives us answers in terms of probabilities: this mean difference is large
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